Analytic gradient for the multireference Brillouin-Wigner coupled cluster method and for the state-universal multireference coupled cluster method

We present the analytic gradient theory and its pilot implementation for the multireference Brillouin-Wigner coupled cluster (BWCC) method and for the state-universal multireference coupled cluster method. The analytic gradient has been derived for three cases: (i) BWCC method without a size-extensivity correction, (ii) BWCC method with the iterative size-extensivity correction, and (iii) for the rigorously size-extensive state-universal method. The pilot implementation is based on full-configuration interaction expansions and is presently limited to single and double excitation levels; however, the resulting equations are general. For BWCC methods, they also do not contain terms explicitly mixing amplitudes of different reference configurations and can thus be implemented in an efficient way. The analytic gradients have been verified with respect to numerically computed ones on the example of CH2 molecule, and geometry optimizations of CH2 and SiH2 have been carried out.     

Multireference second-order perturbation theory: how size consistent is "almost size consistent"

A systematic study of the deviation from size consistency of the multireference second-order Moller-Plesset perturbation theory (MRMP2) method is presented. The size-consistency error is shown to depend on the number of monomers in a supermolecule calculation, size of basis set, number of correlated valence electrons, and size of active space. HF, F(2), and N(2) are used as test cases, with stretched bonds, to include simple, well-defined multireference character. This is essential in ensuring that MRMP2 is being tested as a multireference method. It is concluded that the MRMP2 and other multireference perturbation theory methods can exhibit significant size-consistency errors, and that the size of the error depends on the manner in which the perturbation theory is implemented.     

Coupling term derivation and general implementation of state-specific multireference coupled cluster theories

Simple closed-form expressions are derived for the "same vacuum" renormalization terms that arise in state-specific multireference coupled cluster (MRCC) theories. Explicit equations are provided for these coupling terms through the triple excitation level of MRCC theory, and a general expression is included for arbitrary-order excitations. The first production-level code (PSIMRCC) for state-specific and rigorously size-extensive Mukherjee multireference coupled cluster singles and doubles (MkCCSD) computations has been written. This code is also capable of evaluating analogous Brillouin-Wigner multireference energies (BWCCSD), including a posteriori size-extensivity corrections. Using correlation-consistent basis sets (cc-pVXZ, X=D,T,Q), MkCCSD and BWCCSD were tested and compared on two classic multireference problems: (1) the dissociation potential curve of molecular fluorine (F(2)) and (2) the structure and vibrational frequencies of ozone. Comparison with experimental data shows that the Mukherjee method is generally superior to the Brillouin-Wigner theory in predicting energies, structures, and vibrational frequencies. Particularly accurate results for F(2) are obtained by applying the MkCCSD method with localized molecular orbitals. Although the MkCCSD theory greatly improves upon single-reference CCSD for the geometric parameters and a(1) vibrational frequencies of ozone, the antisymmetric stretching frequency omega(3)(b(2)) remains pathological and cannot be properly treated without the inclusion of connected triple excitations. Finally, preliminary multireference MkCCSD results are reported for the singlet-triplet splittings in ortho-, meta-, and para-benzyne, coming within 1.5 kcal mol(-1) of experiment in all cases.     

Multireference perturbation theory with optimized partitioning. II. Applications to molecular systems

The second‐order multireference perturbation theory using an optimized partitioning, denoted as MROPT(2), is applied to calculations of various molecular properties—excitation energies, spectroscopic parameters, and potential energy curves—for five molecules: ethylene, butadiene, benzene, N₂, and O₂. The calculated results are compared with those obtained with second‐ and third‐order multireference perturbation theory using the traditional partitioning techniques. We also give results from computations using the multireference configuration interaction (MRCI) method. The presented results show very close resemblance between the new method and MRCI with renormalized Davidson correction. The accuracy of the new method is good and is comparable to that of second‐order multireference perturbation theory using Møller‐Plesset partitioning. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1390–1400, 2003     

Binding in transition metal complexes: Reduced multireference coupled-cluster study of the MCH2+ (M=Sc to Cu) compounds

The recently developed reduced multireference coupled-cluster method with singles and doubles (RMR CCSD), which is perturbatively corrected for triples [RMR CCSD(T)], is employed to compute binding energies of nine transition metal ions with CH2. Unlike analogous compounds involving main-group elements, the MCH2+ (M=Sc to Cu) transition metal complexes often exhibit a non-negligible multireference character. The authors thus employ the RMR CCSD(T) method, which represents an extension of the standard single-reference (SR) CCSD(T) method and can account for multireference effects, while employing only small reference spaces. In this way the role of quasidegeneracy effects on the binding energies of these complexes can be assessed at a higher SD(T) level than is possible with the widely used ab initio methods, namely, with the standard SR CCSD(T) approach, and provide a new benchmark for these quantities. The difference between the RMR and the standard CCSD(T) methods becomes particularly evident when considering nonequilibrium geometries.     

Strongly correlated mechanisms of a photoexcited radical reaction from the anti-Hermitian contracted Schrödinger equation

Photoexcited radical reactions are critical to processes in both nature and materials, and yet they can be challenging for electronic structure methods due to the presence of strong electron correlation. Reduced-density-matrix (RDM) methods, based on solving the anti-Hermitian contracted Schro?dinger equation (ACSE) for the two-electron RDM (2-RDM), are examined for studying the strongly correlated mechanisms of these reactions with application to the electrocyclic interconversion of allyl and cyclopropyl radicals. We combine recent extensions of the ACSE to excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 80, 022507 (2009)] and arbitrary spin states [A. E. Rothman, J. J. Foley IV, and D. A. Mazziotti, Phys. Rev. A 80, 052508 (2009)]. The ACSE predicts that the ground-state ring closure of the allyl radical has a high 52.5 kcal/mol activation energy that is consistent with experimental data, while the closure of an excited allyl radical can occur by disrotatory and conrotatory pathways whose transition states are essentially barrierless. Comparisons are made with multireference second- and third-order perturbation theories and multireference configuration interaction. While predicted energy differences do not vary greatly between methods, the ACSE appears to improve these differences when they involve a strongly and a weakly correlated radical by capturing a greater share of single-reference correlation that increases the stability of the weakly correlated radicals. For example, the ACSE predicts a -39.6 kcal/mol conversion of the excited allyl radical to the ground-state cyclopropyl radical in comparison to the -32.6 to -37.3 kcal/mol conversions predicted by multireference methods. In addition, the ACSE reduces the computational scaling with the number of strongly correlated orbitals from exponential (traditional multireference methods) to quadratic. Computed ground- and excited-state 2-RDMs are nearly N-representable.     

The X1Sigmag+, B1Deltag, and B' 1Sigmag+ states of C2: a comparison of renormalized coupled-cluster and multireference methods with full configuration interaction benchmarks

Unusual bonding and electronic near degeneracies make the lowest-lying singlet states of the C2 molecule particularly challenging for electronic structure theory. Here we compare two alternative approaches to modeling bond-breaking reactions and excited states: sophisticated multireference configuration interaction and multireference perturbation theory methods, and a more "black box," single-reference approach, the completely renormalized coupled-cluster method. These approximate methods are assessed in light of their ability to reproduce the full configuration interaction potential energy curves for the X1Sigmag+, B1Deltag, and B' 1Sigmag+ states of C2, which are numerically exact solutions of the electronic Schrodinger equation within the space spanned by a 6-31G* basis set. Both the multireference methods and the completely renormalized coupled-cluster approach provide dramatic improvements over the standard single-reference methods. The multireference methods are nearly as reliable for this challenging test case as for simpler reactions which break only single bonds. The completely renormalized coupled-cluster approach has difficulty for large internuclear separations R in this case, but over the wide range of R=1.0-2.0 A, it compares favorably with the more complicated multireference methods.     

基于完全活性空间自洽场的杂化多组态密度泛函方法λ-DFCAS

提出了一种杂化多组态密度泛函新方法——λ-DFCAS. 在λ-DFCAS方法中, 电子相关能被分为静态和动态相关能. 静态相关能由多组态波函数方法完全活性空间自洽场(CASSCF)得到, 而动态相关能由密度泛函理论方法描述. 两种相关能的杂化比例由一个可调节的参数λ控制. 参数λ的取值取决于分子体系的多组态特性, 在0~1之间变化, 从而使得λ-DFCAS可以应用于各种强相关分子体系. 该方法能够以与CASSCF相当的计算代价获得接近完全活性空间二阶微扰(CASPT2)的计算精度, 并具备了大小一致性.     

Influence of multi-atom bridging ligands on the electronic structure and magnetic properties of homodinuclear titanium molecules

The electronic structure and magnetic properties of homodinuclear titanium(III) molecules with bridging ligands from groups 14, 15, and 16 are examined. Single- and multireference methods with triple-zeta plus polarization basis sets are employed. Dynamic electron correlation effects are included via second-order multireference perturbation theory. Isotropic interaction parameters are calculated, and two of the complexes studied are predicted to be ferromagnetic based on multireference second-order perturbation (MRMP2) theory, using the TZVP(fg) basis set. Zero-field splitting parameters are determined using spin-orbit coupling obtained from complete active space (CAS) self-consistent field (SCF) and multiconfigurational quasi-degenerate perturbation theory (MCQDPT) wave functions. Three Breit-Pauli-based spin coupling methods were employed: full Breit-Pauli (HSO2), the partial two-electron method (P2E), and the semiempirical one-electron method (HSO1).     

The singlet-triplet gap in trimethylenmethane and the ring-opening of methylenecyclopropane: a multireference Brillouin-Wigner coupled cluster study

We performed an ab initio study of the singlet-triplet gap in trimethylenmethane (TMM) and of the ring-opening of methylenecyclopropane by the multireference BWCC method. Since the singlet states of TMM and intermediates between TMM and methylenecyclopropane have a strong multiconfigurational character, it is necessary to use a multireference method. The cc-pVDZ and cc-pVTZ basis sets were used. We compared our results with experiments, where available, and with previous calculations performed by MCSCF and spin-flip coupled-cluster-type methods.     

Optimizing conical intersections without derivative coupling vectors: application to multistate multireference second-order perturbation theory (MS-CASPT2)

We introduce a new method for optimizing minimal energy conical intersections (MECIs), based on a sequential penalty constrained optimization in conjunction with a smoothing function. The method is applied to optimize MECI geometries using the multistate formulation of second-order multireference perturbation theory (MS-CASPT2). Resulting geometries and energetics for conjugated molecules including ethylene, butadiene, stilbene, and the green fluorescent protein chromophore are compared with state-averaged complete active space self-consistent field (SA-CASSCF) and, where possible, benchmark multireference single- and double-excitation configuration interaction (MRSDCI) optimizations. Finally, we introduce the idea of "minimal distance conical intersections", which are points on the intersection seam that lie closest to some specified geometry such as the Franck-Condon point or a local minimum on the excited state.     

Calculation of excitation energies of open-shell molecules with spatially degenerate ground states. I. Transformed reference via an intermediate configuration Kohn-Sham density-functional theory and applications to d1 and d2 systems with octahedral and tetrahedral symmetries

A method for calculating the UV-vis spectra of molecules with spatially degenerate ground states using time-dependent density-functional theory (TDDFT) is proposed. The new transformed reference via an intermediate configuration Kohn-Sham TDDFT (TRICKS-TDDFT) method avoids the difficulties caused by the multireference nature of spatially degenerate states by rather than utilizing the ground state instead taking a nondegenerate excited state with desirable properties as the reference for the TDDFT calculation. The scope and practical application of the method are discussed. Like all open-shell TDDFT calculations this method at times suffers from the inability to produce transitions to states that are eigenfunctions of the total spin operator. A technique for alleviating this difficulty to some extent is proposed. The applicability and accuracy of the TRICKS-TDDFT method is demonstrated through example calculations of several d(1) and d(2) transition metal complexes with tetrahedral and octahedral symmetries. For the most part, the results of these calculations are similar in quality to to those obtained from standard TDDFT calculations.     

Reduced multireference coupled-cluster method: barrier heights for heavy atom transfer, nucleophilic substitution, association, and unimolecular reactions

The recently developed reduced multireference coupled-cluster method with singles and doubles (RMR CCSD) that is perturtatively corrected for triples [RMR CCSD(T)] is employed to compute the forward and reverse barrier heights for 19 non-hydrogen-transfer reactions. The method represents an extension of the conventional single-reference (SR) CCSD(T) method to multireference situations. The results are compared with a benchmark database, which is essentially based on the SR CCSD(T) results. With the exception of seven cases, the RMR CCSD(T) results are almost identical with those based on SR CCSD(T), implying the abatement of MR effects at the SD(T) level relative to the SD level. Using the differences between the RMR CCSD(T) and CCSD(T) barrier heights as a measure of MR effects, modified values for barrier heights of studied reactions are given.     

Simplified reference wave functions for multireference perturbation theory

A number of simplifications in defining the reference wave functions used in multireference second-order M?ller-Plesset perturbation theory (MRMP2) calculations are studied. The usual multiconfigurational orbital optimization is avoided by using Hartree-Fock or Kohn-Sham orbitals; the complete configuration expansion in the active-space orbitals is replaced by a severely truncated expansion, and the spin-component-scaling idea is applied to the multireference perturbation expansion. We assess these approximations to the full procedure by calculating the barrier heights for 15 processes taken from the Zhao-Gonzalez-Garcia-Truhlar database. Our results suggest that reliable and relatively cheap reference wave functions for MRMP2 calculations can be obtained from the simplifications introduced here. We hope that this will enable the application of the MRMP2 method to a larger range of chemical systems.     

Linear-response theory for Mukherjee's multireference coupled-cluster method: Excitation energies

The recently presented linear-response function for Mukherjee's multireference coupled-cluster method (Mk-MRCC) [T.-C. Jagau and J. Gauss, J. Chem. Phys. 137, 044115 (2012)] is employed to determine vertical excitation energies within the singles and doubles approximation (Mk-MRCCSD-LR) for ozone as well as for o-benzyne, m-benzyne, and p-benzyne, which display increasing multireference character in their ground states. In order to assess the impact of a multireference ground-state wavefunction on excitation energies, we compare all our results to those obtained at the single-reference coupled-cluster level of theory within the singles and doubles as well as within the singles, doubles, and triples approximation. Special attention is paid to the artificial splitting of certain excited states which arises from the redundancy intrinsic to Mk-MRCC theory and hinders the straightforward application of the Mk-MRCC-LR method.     

Uncoupled multireference state-specific Mukherjee's coupled cluster method with triexcitations

We have developed the uncoupled version of multireference Mukherjee's coupled cluster method with connected triexcitations. The method has been implemented in ACES II program package. The agreement between the uncoupled and the standard version of Mukherjee's multireference coupled cluster method has been reported previously at the singles and doubles level by Das et al. [J. Mol. Struct.: THEOCHEM 79, 771 (2006); Chem. Phys. 349, 115 (2008)]. The aim of this article is to investigate this method further, in order to establish how its performance changes with the size of the basis set, size of the model space, multireference character of different molecules, and inclusion of connected triple excitations. Assessment of the new method has been performed on the singlet methylene, potential energy curve of fluorine molecule, and third b?(1)Σ(g)(+) electronic state of oxygen molecule.